John wiley sons fundamentals of liquid crystal devices spy

The flexible tails are alsonecessary, otherwise the molecules form the crystal phase where there is positional order.The variety of phases that may be exhibited by rod-like molecules are

Crystal Devices

Fundamentals of Liquid Crystal Devices D.-K Yang and S.-T Wu

# 2006 John Wiley & Sons, Ltd ISBN: 0-470-01542-X

Design and Applications

Lindsay W Macdonaldand Anthony C Lowe (Eds)

Electronic Display Measurement:

Concepts, Techniques and Instrumentation

Peter A Keller

Projection Displays

Edward H Stuppand Matthew S Brennesholz

Liquid Crystal Displays:

Addressing Schemes and Electro-Optical Effects

Ernst Lueder

Reflective Liquid Crystal Displays

Shin-Tson Wuand Deng-Ke Yang

Colour Engineering:

Achieving Device Independent Colour

Phil Greenand Lindsay MacDonald (Eds)

Display Interfaces:

Fundamentals and Standards

Robert L Myers

Digital Image Display:

Algorithms and Implementation

Gheorghe Berbecel

Flexible Flat Panel Displays

Gregory Crawford (Ed.)

Polarization Engineering for LCD Projection

Michel G Robinson, Jianmin Chen, and Gary D Sharp

Fundamentals of Liquid Crystal Devices

Deng-Ke Yangand Shin-Tson Wu

Fundamentals of Liquid Crystal Devices

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Library of Congress Cataloging-in-Publication Data

Yang, Deng-Ke.

Fundamentals of liquid crystal devices / Deng-Ke Yang, Shin-Tson Wu.

p cm.

Includes bibliographical references and index.

ISBN 0-470-01542-X (cloth : alk paper)

1 Liquid crystal devices –Textbooks 2 Liquid crystal displays–Textbooks 3 Liquid crystals–Textbooks.

I Wu, Shin-Tson II Title.

TS518.Y36 2006

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN-13 978-0-470-01542-1 (HB)

ISBN-10 0-470-01542-X (HB)

Typeset in 9/11pt Times by Thomson Digital Noida.

Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire

This book is printed on acid-free paper responsibly manufactured from sustainable forestry

in which at least two trees are planted for each one used for paper production.

1.3.2 Landau–de Gennes theory of orientational order in

1.4.2 Elastic properties of cholesteric liquid

1.6 Anchoring Effects of Nematic Liquid Crystals at

2 Propagation of Light in Anisotropic Optical Media 39

2.2.1 Monochromatic plane waves and their

3.2.5 Evolution of the polarization states on the Poincare´

4.3.3 Surface-stabilized ferroelectric liquid crystals 1204.3.4 Electroclinic effect in chiral smectic- liquid crystals 122

5.3.1 Dynamics of the Freedericksz transition in twist geometry 144

7.2 Modeling the Electric Field 184

8.5.2 MVA with a positive a, a negative a, and a negative c film 231

I was first exposed to liquid crystal displays in 1972 when I joined Westinghouse R&D Center as aresearch engineer Later, I was assigned the task of developing a 500 500100 100 pixels TFT-LCDpanel for the US Air Force At that time, the LCD fabrication processes were very primitive The LCalignment was an obliquely evaporated SiOx film The cell spacing was done by mylar strips for thesecond minimum cell gap around 8 microns The filling of LC was in a vacuum chamber with ahypodermic syringe holding the LC and a needle glued by Epoxy to the covering glass through a holedrilled at the corner of the monochrome ITO covering glass The LC filling could take half a day Whatgiant progress the LC fabrication processes have made over the past 30 years The sizes of the glasssubstrates can reach almost 4 square meters The uniform cell gap of LC first minimum around 3–4microns is maintained by photopatterned spacers Using the one-drop-fill technique the LC filling can

be done in just a few minutes for a 4200-diagonal panel

During the last 30 some years, the applications of LCDs have expanded tremendously Startingfrom watches and calculators in the1970s with direct-drive TNs, the portable PC terminals utilizedmonochrome simple-matrix-addressed TNs in the early 1980s, which were followed by monochromeSTNs In the 1990s, dual-scan and multiline addressed STN color panels for notebook PCs appeared Inthe early 1990s, the mass-produced color TFT-LCD TN notebook PC panels started the unlimiteddevelopmental changes by TFT-LCD panels for increasingly larger panels for notebook PCs, monitorsand eventually TV applications Today, TFT-LCD TV panels have been sold in the market with sizes

up to 5000and performances comparable to CRTs TFT-LCDs have established their position as thedominating flat panel display technology Nevertheless, along the long developmental paths, there havebeen numerous challengers such as thin film EL, vacuum fluorescence, electrochromics, PDP and FED.Each time, LCDs have been able to sustain the challenges and emerge victoriously Now, it lookscertain that TFT-LCDs will replace the ultimate king, CRT, in the home’s living room as the primaryvideo display

The LCD technologies have amazing resources and versatility For each new application, LCDengineers have been able to expand the LCD capabilities to meet the new demands in performance.Taking the latest TV application as an example, the LCD engineers have been able to solve issues such

as wide viewing angle, fast response, color shift versus gray levels and viewing angles

What will be the next major application beyond TV? We don’t know yet However, we are certainthat LCD engineers will be able to rise above the challenges and bring the technology to the next level.The LCD technologies will be the dominating flat panel display for many years to come

There have been many books related to LCD technologies This book authored by Professor Yangand Professor Wu is the most extensive textbook on LCD which covers the physics, materials anddriving methods of many types of LCD modes I’m sure this book will provide students entering theLCD field with a thorough understanding of the devices.

Fan LuoVice President and CTO

AU OptronicsScience-Based Industrial Park

HsinchuTaiwan

Series Editor’s Foreword

Since its inception, the purpose of this series has been to provide works for practising scientists andengineers in the field of display technology This book will most certainly fulfil that purpose, but it willalso mark a new departure for the series as it has been written primarily as a text for postgraduate andsenior undergraduate students This broadening of the aims of the series is welcome since itacknowledges the need to provide works in the display field not only to those already working inthe profession but also to those who are about to enter it

Liquid crystal display technology has developed enormously over the past 35 years It is alreadydominating many display markets such as hand held PDA and phone products, notebook computerdisplays and desktop monitors In those markets where it does not have total domination, such as flat

TV, it has a very healthy and increasing market share Moreover, although twisted and supertwistednematic technologies were for many years ubiquitous, as performance requirements have become morestringent, other effects such as in-plane switching and vertically aligned nematics have come to the foreand new effects continue to be invented For the very lowest power reflective displays, cholesteric and

a variety of bistable nematic effects are used And so far, I have only discussed direct view displays!

LC projection displays, which can use only a single polarizer, require extremely innovative twisteffects With so many different liquid crystal effects being used so widely in industry, the addition ofthis latest book to the series is very timely

Designed to be self-contained, the first chapters cover the basic physics of liquid crystals, theirinteraction with light and electric fields and the means by which they can be modelled Next aredescribed the majority of ways in which liquid crystals can be used in displays and a final chapter dealswith photonic devices such as beam steerers, tunable focus lenses and polarisation-independentdevices

Because it is intended to be used as a textbook, another innovation has been introduced for the firsttime in the series Each chapter concludes with a set of problems, the answers to which may be found

on the Wiley web site

Written by two academics of world standing in their fields – Deng-Ke Yang is a specialist incholesterics and polymer stabilised systems and Shin-Tson Wu has made major contributions in twistsystems for reflective displays and in many other areas – this latest book is a most welcome addition tothe series

Anthony C LoweBraishfield, UK 2006

Liquid crystal displays have become the leading technology in the information display industry Theyare used in small-sized displays such as calculators, cellular phones, digital cameras, and head-mounted displays; in medium-sized displays such as laptop and desktop computers; and in large-sizeddisplays such as direct-view TVs and projection TVs They have the advantages of high resolution andhigh brightness, and, being flat paneled, are lightweight, energy saving, and even flexible in somecases They can be operated in transmissive and reflective modes Liquid crystals have also been used

in photonic devices such as laser beam steering, variable optical attenuators, and tunable-focus lenses.There is no doubt that liquid crystals will continue to play an important role in the era of informationtechnology

There are many books on the physics and chemistry of liquid crystals or on liquid crystal devices.There are, however, only few books covering both the basics and applications of liquid crystals Themain goal of this book is to provide a textbook for senior undergraduate and graduate students Thebook can be used for a one- or two-semester course The instructors can selectively choose the chaptersand sections according to the length of the course and the interest of the students The book can also beused as a reference book by scientists and engineers who are interested in liquid crystal displays andphotonics

The book is organized in such a way that the first few chapters cover the basics of liquid crystalsand the necessary techniques to study and design liquid crystal devices The later chapters cover theprinciples, design, operation, and performance of liquid crystal devices Because of limited space, wecannot cover every aspect of liquid crystal chemistry and physics and all liquid crystal devices, but wehope this book can introduce readers to liquid crystals and provide them with the basic knowledge andtechniques for their careers in liquid crystals

We are greatly indebted to Dr A Lowe for his encouragement We are also grateful to thereviewers of our book proposal for their useful suggestions and comments Deng-Ke Yang would like

to thank Ms E Landry and Prof J Kelly for patiently proof-reading his manuscript He would alsolike to thank Dr Q Li for providing drawings Shin-Tson Wu would like to thank his research groupmembers for generating the new knowledge included in this book, especially Drs Xinyu Zhu,Hongwen Ren, Yun-Hsing Fan, and Yi-Hsin Lin, and Mr Zhibing Ge for kind help during manuscriptpreparation He is also indebted to Dr Terry Dorschner of Raytheon, Dr Paul McManamon of the AirForce Research Lab, and Dr Hiroyuki Mori of Fuji Photo Film for sharing their latest results Wewould like to thank our colleagues and friends for useful discussions and drawings and our funding

agencies (DARPA, AFOSR, AFRL, and Toppoly) for providing financial support Finally, we alsowould like to thank our families (Xiaojiang Li, Kevin Yang, Steven Yang, Cho-Yan Wu, Janet Wu, andBenjamin Wu) for their spiritual support, understanding, and constant encouragement.

Deng-Ke YangShin-Tson Wu

of the rod-like molecules or the ratio between the diameter and the thickness of the disk-like molecules isabout 5 or larger Because the molecules are non-spherical, besides positional order, they may alsopossess orientational order.

Figure 1.1(a) shows a typical calamitic liquid crystal molecule Its chemical name is 40cyano-biphenyl and is abbreviated as 5CB [4,5] It consists of a biphenyl, which is the rigid core, and ahydrocarbon chain, which is the flexible tail The space-filling model of the molecule is shown inFigure 1.1(c) Although the molecule itself is not cylindrical, it can be regarded as a cylinder, as shownFigure 1.1(e), in considering its physical behavior because of the fast rotation (on the order of 109s)around the long molecular axis due to thermal motion The distance between two carbon atoms isabout1:5 A8; therefore the length and the diameter of the molecule are about 2 nm and 0.5 nm,respectively The molecule shown has a permanent dipole moment (from the CN head); however, itcan still be represented by a cylinder whose head and tail are the same, because in non-ferroelectricliquid crystal phases, the dipole has equal probability of pointing up or down It is necessary for a liquidcrystal molecule to have a rigid core(s) and flexible tail(s) If the molecule is completely flexible, it willnot have orientational order If it is completely rigid, it will transform directly from the isotropic liquidphase at high temperature to the crystalline solid phase at low temperature The rigid part favors bothorientational and positional order while the flexible part does not With balanced rigid and flexible parts,the molecule exhibits liquid crystal phases

-n-pentyl-4-Figure 1.1(b) shows a typical discotic liquid crystal molecule [6] It also has a rigid core and flexibletails The branches are approximately on one plane The space-filling model of the molecule is shown

in Figure 1.1(d) If there is no permanent dipole moment perpendicular to the plane of the molecule, itcan be regarded as a disk in considering its physical behavior as shown in Figure 1.1(f), because of thefast rotation around the axis which is at the center of the molecule and perpendicular to the plane of themolecule If there is a permanent dipole moment perpendicular to the plane of the molecule, it is better to

Fundamentals of Liquid Crystal Devices D.-K Yang and S.-T Wu

# 2006 John Wiley & Sons, Ltd ISBN: 0-470-01542-X

visualize the molecule as a bowl, because the reflection symmetry is broken and all the permanentdipoles may point in the same direction and spontaneous polarization occurs The flexible tails are alsonecessary, otherwise the molecules form the crystal phase where there is positional order.

The variety of phases that may be exhibited by rod-like molecules are shown in Figure 1.2 At hightemperature, the molecules are in the isotropic liquid state where they do not have either positional ororientational order The molecules can easily move around and the material can flow like water Thetranslational viscosity is comparable to that of water Both the long and short axes of the molecules canpoint in any direction

When the temperature is decreased, the material transforms into the nematic phase, which is the mostcommon and simplest liquid crystal phase, where the molecules have orientational order but still nopositional order The molecules can still diffuse around and the translational viscosity does not changemuch from that of the isotropic liquid state The long axis of the molecules has preferred direction.Although the molecules still swivel due to thermal motion, the time-averaged direction of the long axis

of a molecule is well defined and is the same for all the molecules at the macroscopic scale The average

(f) (e)

L ~ 2 nm

D ~ 0.5 nm

Figure 1.1 Calamitic liquid crystal: (a) chemical structure, (c) space-filling model, (e) physicalmodel Discostic liquid crystal: (b) chemical structure, (d) space-filling mode, (f) physical model

direction of the long molecular axis is denoted by ~n which is a unit vector and called the liquid crystaldirector The short axes of the molecules have no orientational order in a uniaxial nematic liquid crystal.When the temperature is decreased further, the material may transform into the smetic-A phase where,besides the orientational order, the molecules have partial positional order, i.e., the molecules form alayered structure The liquid crystal director is perpendicular to the layers Smectic-A is a one-dimensional crystal where the molecules have positional order in the normal direction of the layer.The diagram shown in Figure 1.2 is schematic In reality, the separation between neighboring layers isnot as well defined as that shown in the figure The molecular number density exhibits an undulation withthe wavelength about the molecular length Within a layer, it is a two-dimensional liquid in which there

is no positional order and the molecules can move around For a material in poly-domain smectic-A, thetranslational viscosity is significantly higher, and it behaves like a grease When the temperature isdecreased still futher, the material may transform into the smectic-C phase where the liquid crystaldirector is no longer perpendicular to the layer but tilted

At low temperature, the material is in the crystal solid phase where there are both positional andorientational orders The translational viscosity become infinite and the molecules (almost) no longerdiffuse

Liquid crystals get the ‘crystal’ part of their name because they exhibit optical birefringence likecrystalline solids They get the ‘liquid’ part of their name because they can flow and do not supportshearing like regular liquids Liquid crystal molecules are elongated and have different molecularpolarizabilities along their long and short axes Once the long axes of the molecules orient along acommon direction, the refractive indices along and perpendicular to the common direction are different

It should be noted that not all rod-like molecules exhibit all the liquid crystal phases, but just some ofthem

Some of the liquid crystal phases of disk-like molecules are shown in Figure 1.3 At high temperature,they are in the isotropic liquid state where there are no positional and orientational orders The materialbehaves in the same way as a regular liquid When the temperature is decreased, the material transforms

Figure 1.2 Schematic representation of the phases of rod-like molecules

Figure 1.3 Schematic representation of the phases of disk-like molecules

into the nematic phase which has orientational order but not positional order The average direction ofthe short axis perpendicular to the disk is oriented along a preferred direction which is also called theliquid crystal director and denoted by a unit vector ~n The molecules have different polarizabilities along

a direction in the plane of the disk and along the short axis Thus the discotic nematic phase also exhibitsbirefringence as crystals

When the temperature is decreased further, the material transforms into the columnar phase where,besides orientational order, there is partial positional order The molecules stack up to form columns.Within a column, the columnar phase is a liquid where the molecules have no positional order.The columns, however, are arranged periodically in the plane perpendicular to the columns Hencethe columnar phase is a two-dimensional crystal At low temperature, the material transforms into thecrystalline solid phase where the positional order along the columns is developed

The liquid crystal phases discussed so far are called thermotropic liquid crystals and the transitionsfrom one phase to another phase are driven by varying temperature There is another type of liquidcrystallinity, called lyotropic, exhibited by molecules when they are mixed with a solvent of some kind.The phase transitions from one phase to another phase are driven by varying the solvent concentration.Lyotropic liquid crystals usually consist of amphiphilic molecules which have a hydrophobic group atone end and a hydrophilic group at the other end, with water as the solvent The common lyotropic liquidcrystal phases are micelle phase and lamellar phase Lyotropic liquid crystals are important in biology.They will not be discussed in this book because its scope concerns displays and photonic devices.Liquid crystals have a history of more than 100 years It is believed that the person who discoveredliquid crystals was Friedrich Reinitzer, an Austrian botanist [7] The liquid crystal phase observed byhim in 1888 was a cholesteric phase Since then liquid crystals have come a long way and become amajor branch of interdisciplinary science Scientifically, liquid crystals are important because of therichness of their structures and transitions Technologically, they have gained tremendous success indisplay and photonic applications [8–10]

1.2 Thermodynamics and Statistical Physics

Liquid crystal physics is an interdisciplinary science, involving thermodynamics, statistical physics,electrodynamics, and optics Here we give a brief introduction to thermodynamics and statisticalphysics

1.2.1 Thermodynamic laws

One of the important quantities in thermodynamics is entropy From the microscopic point of view,entropy is a measure of the number of quantum states accessible to a system In order to define entropyquantitatively, we first consider the fundamental logical assumption that for a closed system (in which noenergy and particles exchange with other systems), quantum states are either accessible or inaccessible

to the system, and the system is equally likely to be in any one of the accessible states as in any otheraccessible state[11] For a macroscopic system, the number of accessible quantum states g is a hugenumber ( 1023) It is easier to deal with ln g, which is defined as the entropy s:

If a closed system consists of subsystem 1 and subsystem 2, the numbers of accessible states of thesubsystems are g1and g2, respectively The number of accessible quantum states of the whole system is

g¼ g1g2 and the entropy is s¼ ln g ¼ ln ðg1g2Þ ¼ ln g1þ ln g2¼ s1þ s2

Entropy is a function of the energy u of the system s¼ sðuÞ The second law of thermodynamicsstates that for a closed system, the equilibrium state has maximum entropy Let us consider a closedsystem which contains two subsystems When two subsystems are brought into thermal contact the

energy exchange between them is allowed, the energy is allocated to maximize the number of accessiblestates; that is, the entropy is maximized Subsystem 1 has energy u1and entropy s1; subsystem 2 hasenergy u2 and entropy s2 For the whole system, u¼ u1þ u2 and s¼ s1þ s2 The first law ofthermodynamics states that energy is conserved, i.e., u¼ u1þ u2¼ constant For any process inside theclosed system, du¼ du1þ du2¼ 0 From the second law of thermodynamics, for any process we have

ds¼ ds1þ ds2 0 When the two subsystems are brought into thermal contact, at the beginning energyflows For example, an amount of energyjdu1j flows from subsystem 1 to subsystem 2, du1<0 and

Now we consider the thermodynamics of a system at a constant temperature, i.e., in thermal contact with

a thermal reservoir The temperature of the thermal reservoir (named B) is t The system underconsideration (named A) has two states with energy 0 and e, respectively A and B form a closed system,and its total energy u¼ uAþ uB¼ uo¼ constant When A is in the state with energy 0, B has energy uo,and the number of accessible states is g1¼ gA B BðuoÞ ¼ gBðuoÞ When A has energy e, B hasenergy uo e, and the number of accessible states is g2¼ gA B Bðuo eÞ ¼ gBðuo eÞ Forthe whole system, the total number of accessible states is

ðA þ BÞ is a closed system, and the probability in any of the G states is the same When the wholesystem is in one of the g states, A has energy 0 When the whole system is in one of the g states, A has

energy e Therefore the probability that A is in the state with energy 0 is

Pð0Þ ¼ g1

g1þ g2

¼ gBðuoÞ

gBðuoÞ þ gBðuo eÞ

The probability that A is in the state energy e is

PðeÞ ¼ g2

g1þ g2¼

gBðuo eÞ

gBðuoÞ þ gBðuo eÞ

From the definition of entropy, we have gBðuoÞ ¼ esBðu o Þand gBðuo eÞ ¼ esBðu o eÞ Because e uo,

sBðuo eÞ sBðuoÞ @ B

eieei =k B T

then

U¼kBT2Z

@Z

@T¼ kBT2@ðln ZÞ

Its variation in a reversible process is given by

dF¼ dU  dðTSÞ ¼ ðTdS  PdVÞ  ðTdS þ SdTÞ ¼ SdT  PdV (1.24)

The physical meaning of Helmholtz free energy is that in a process at constant temperature, the change

of Helmholtz free energy is equal to the work done to the system The derivatives are

1.2.4 Criteria for thermodynamic equilibrium

Now we consider the criteria which can used to judge whether a system is in its equilibrium state undergiven conditions We already know that for a closed system, as it changes from a non-equilibrium state tothe equilibrium state, the entropy increases:

It can be stated differently that, for a closed system, the entropy is maximized in the equilibrium state

In considering the equilibrium state of a system at constant temperature and volume, we construct aclosed system which consists of the system (subsystem 1) under consideration and a thermal reservoir(subsystem 2) with temperature T When the two systems are brought into thermal contact, energy isexchanged between subsystem 1 and subsystem 2 Because the whole system is a closed system,

dS¼ dS1þ dS2 0 For system 2, 1=T ¼ ð@S2=@U2ÞV, and therefore dS2¼ dU2=T (this is true whenthe volume of the subsystem is fixed, which also means the volume of subsystem 1 is fixed) Because ofenergy conservation, dU2¼ dU1 Hence dS¼ dS1þ dS2¼ dS1þ dU2= ¼ dS1 dU1=  0.Because the temperature and volume are constant for subsystem 1, dS1 dU1= ¼ ð1=T ÞdðTS1 U Þ  0, and therefore

PV1Þ  0 Therefore

At constant temperature and pressure, the equilibrium state has minimum Gibbs free energy Ifelectric energy is involved, then we have to consider the electric work done to the system by externalsources such as a battery In a thermodynamic process, if the electric work done to the system is dWe,then

1.3 Orientational Order

Orientational order is the most important feature of liquid crystals The average directions of the longaxes of the rod-like molecules are parallel to each other Because of the orientational order, liquidcrystals possess anisotropic physical properties; that is, in different directions, they have differentresponses to external fields such as an electric field, a magnetic field, and shear In this section, we willdiscuss how to quantitatively specify orientational order and why rod-like molecules tend to be parallel

to each other

For a rigid elongated liquid crystal molecule, three axes can be attached to it to describe its orientation.One is the long molecular axis and the other two are perpendicular to the long molecular axis Usually themolecule rotates rapidly around the long molecular axis Although the molecule is not cylindrical, ifthere is no hindrance to the rotation in the nematic phase, the rapid rotation around the long molecularaxis makes it behave like a cylinder There is no preferred direction for the short axes and thus thenematic liquid crystal is usually uniaxial If there is hindrance to the rotation, the liquid crystal is biaxial

A biaxial nematic liquid crystal is a long sought for material The lyotropic biaxial nematic phase hasbeen observed [12] The existence of a thermotropic biaxial nematic phase is still under debate, and itmay exist in bent-core molecules [13,14] Here our discussion is on bulk liquid crystals The rotationalsymmetry around the long molecular axis can be broken by confinement In this book, we will deal withuniaxial liquid crystals consisting of rod-like molecules unless otherwise stated

1.3.1 Orientational order parameter

In uniaxial liquid crystals, we have only to consider the orientation of the long molecular axis Theorientation of a rod-like molecule can be represented by a unit vector ˆa which is attached to the moleculeand parallel to the long molecular axis In the nematic phase, the average directions of the long molecularaxes are along a common direction: namely, the liquid crystal director denoted by the unit vector ~n The3-D orientation of ˆa can be specified by the polar angle y and the azimuthal angle f where the z axis ischosen parallel to ~n as shown in Figure 1.4 In general the orientational order of ˆa is specified by anorientational distribution function fðy; fÞ f ðy; fÞdOðdO ¼ sin ydydfÞ is the probability that ˆa isoriented along the direction specified by y and f within the solid angle dO In the isotropic phase, ˆa hasequal probability of pointing in any direction and therefore fðy; fÞ ¼ constant For uniaxial liquidcrystals, there is no preferred orientation in the azimuthal direction, and then f ¼ f ðyÞ which dependsonly on the polar angle y

Rod-like liquid crystal molecules may have permanent dipole moments If the dipole moment isperpendicular to the long molecular axis, the dipole has equal probability of pointing along any directionbecause of the rapid rotation around the long molecular axis in uniaxial liquid crystal phases The dipoles

of the molecules cannot generate spontaneous polarization If the permanent dipole moment is along the

Figure 1.4 Schematic diagram showing the orientation of rod-like molecules

long molecular axis, the flip of the long molecular axis is much slower (of the order of 105s), so theabove argument does not hold In order to see the orientation of the dipoles in this case, we consider theinteraction between two dipoles [15] When one dipole is on top of the other, if they are parallel, theinteraction energy is low and thus parallel orientation is preferred When two dipoles are side by side, ifthey are anti-parallel, the interaction energy is low and thus anti-parallel orientation is preferred As weknow, the molecules cannot penetrate each other For elongated molecules, the distance between twodipoles when they are on top of each other is farther than that when they are side by side The interactionenergy between two dipoles is inversely proportional to the cubic power of the distance between them.Therefore anti-parallel orientation of dipoles is dominant in rod-like molecules There are the samenumber of dipoles aligned parallel to the liquid crystal director ~n as there are aligned anti-parallel to ~n.The permanent dipole along the long molecular axis cannot generate spontaneous polarization Thus,even when the molecules have a permanent dipole moment along the long molecular axes, they can beregarded as cylinders whose top and bottom are the same It can also be concluded that ~n and~n areequivalent.

An order parameter must be defined in order to quantitatively specify the orientational order Theorder parameter is usually defined in such a way that it is zero in the high-temperature unordered phaseand non-zero in the low-temperature ordered phase By analogy with ferromagnetism, we may considerthe average value of the projection of ˆa along the director ~n, i.e.,

hcosyi ¼

Zp 0cos yfðyÞ sin ydy

Zp 0

wherehi indicate the average (the temporal and spatial averages are the same) and cos y is the first-orderLegendre polynomial In the isotropic phase, the molecules are randomly oriented andhcosyi ¼ 0 Wealso know that in the nematic phase the probability that a molecule will orient at angles y and p y isthe same, i.e., fðyÞ ¼ f ðp  yÞ; therefore hcosyi ¼ 0, and so hcosyi provides no information about theorientational order parameter Next, let us consider the average value of the second-order Legendrepolynomial for the order parameter:

S¼ hP2ð cos yÞi ¼

1

2ð3 cos2y 1Þ ¼

Zp 0

1

2ð3 cos2y1Þ f ðyÞ sin ydy

Zp 0fðyÞ sin ydy (1.38)

In the isotropic phase as shown in Figure 1.5(b), fðyÞ ¼ c, a constant, and

1

2ð3 cos2y 1Þc sin ydy ¼ 0

In the nematic phase, fðyÞ depends on y For a perfectly ordered nematic phase as shown inFigure 1.5(d), fðyÞ ¼ dðyÞ, where sin ydðyÞ ¼ 1 when y ¼ 0, sin ydðyÞ ¼ 0 when y 6¼ 0, and

Rp

0dðyÞsin ydy ¼ 1, and the order parameter is S ¼1ð3cos2y 1Þ ¼ 1 It should be pointed out thatthe order parameter can be positive or negative Two order parameters with the same absolute value butdifferent signs correspond to different states When the molecules all lie in a plane but are randomlyoriented in the plane as shown in Figure 1.5(a), the distribution function is fðyÞ ¼ dðy  p=2Þ, wheredðy  p=2Þ ¼ 1 when y ¼ p=2, dðy  p=2Þ ¼ 0 when y 6¼ p=2, andRp

0dðy  p=2Þsin ydy ¼ 1, and theorder parameter is S¼1½3cos2ðp=2Þ  1Þ=1 ¼ 0:5 In this case, the average direction of the molecules

is not well defined The director ~n is defined by the direction of the uniaxial axis of the material.Figure 1.5(c) shows the state with the distribution function fðyÞ ¼ ð35=16Þ½cos4yþ ð1=35Þ, which isplotted vs y in Figure 1.5(e) The order parameter is S¼ 0:5 Many anisotropies of physical propertiesare related to the order parameter and will be discussed later.

1.3.2 Landau–de Gennes theory of orientational order in the nematic phase

Landau developed a theory for second-order phase transitions [16], such as those from the diamagneticphase to the ferromagnetic phase, in which the order parameter increases continuously from zero as thetemperature is decreased across the transition temperature Tcfrom the high-temperature disorderedphase to the low temperature ordered phase For a temperature near Tc, the order is very small The freeenergy of the system can be expanded in terms of the order parameter

The transition from water to ice at 1 atmosphere pressure is a first-order transition and the latentheat is about 100 J=g The isotropic–nematic transition is a weak first-order transition because theorder parameter changes discontinuously across the transition but the latent heat is only about 10 J=g

De Gennes extended Landau’s theory to the isotropic–nematic transition because it is a weak first-ordertransition [1, 17] The free energy density f of the material can be expressed in terms of the orderparameter S as

a given value ofjmj, there is only one state, and the sign of m is decided by the choice of the coordinate.The free energy must be the same for a positive m and a negative m, and therefore the coefficient of thecubic term must be zero For nematic liquid crystals, positive and negative values of the order parameter

S correspond to two different states and the corresponding free energies can be different, and therefore b

is not zero; b must be positive because at sufficiently low temperatures positive order parameters have

f (q )

0.0 0.5 1.0 1.5 2.0 2.5

1.0 0.8 0.6 0.4 0.2 0.0

global minimum free energies We also know that the maximum value of S is 1 The quadratic term with

a positive c prevents S from exploding The values of the coefficients can be estimated in the followingway: the energy of the intermolecular interaction between the molecules associated with orientation isabout 0:1 eV and the molecular size is about 1 nm, f is the energy per unit volume, and therefore

temperature, the order parameter S is found by minimizing f :

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b2 4acðT  TÞq

S3¼ 12c b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b2 4acðT  TÞq

S1¼ 0 corresponds to the isotropic phase and the free energy is f1¼ 0 The isotropic phase has globalminimum free energy at a high temperature It will be shown that at a low temperature S2has globalminimum free energy

S3has a local minimum free energy At the nematic–isotropic phase transition temperature TNI, the orderparameter is Sc¼ S2c, and f2ðS2¼ ScÞ ¼ f1¼ 0; that is,

and the order parameter at the transition temperature

Sc¼2b

For liquid crystal 5CB, the experimentally measured order parameter is shown by the solid circles inFigure 1.6(a) [6] In fitting the data, the following parameters were used: a¼ 0:023s J=Km3, b¼1:2s J=m3;and c¼ 2:2s J=m3, where s is a constant which has to be determined by the latent heat of thenematic–isotropic transition

Because S is a real number in the region from 0:5 to 1.0, when T  T>b2=4ac, i.e., when

T TNI>b2=4ac 2b2=9ac¼ b2=36ac, S2and S3are not real The only real solution is S¼ S1¼ 0,corresponding to the isotropic phase When T TNI <b2=36ac, there are three solutions However,when 0 < T TNI  b2=36ac, the isotropic phase is the stable state because its free energy is still theglobal minimum as shown in Figure 1.6(b) When T TNI  0, the nematic phase with order parameter

0.7 0.6 0.5 0.4 0.3 0.2

0.2 0.1

0.1 0.0

0.0 –0.1

–0.1 –0.2 –0.3 –0.4 –0.5

transition takes place and the order parameter changes discontinuously from 0 to Sc¼ 2b=3c This is afirst-order transition It can be seen from the figure that at this temperature there is an energy barrierbetween S1and S2 If the system is initially in the isotropic phase and there are no means to overcome theenergy barrier, it will remain in the isotropic phase at this temperature As the temperature is decreased,the energy barrier is lowered At T5¼ TNI 3C, the energy barrier is low At T6¼ T, the second-order derivative of f with respect to S at S1¼ 0 is

@2f

@S2



S¼0

¼ aðT  TÞ  2bS þ 3cS2



S¼0

T2(superheating temperature) at which the nematic phase becomes absolutely unstable can be foundfrom

, we can get T2¼ TNIþ b2=36ac

In reality, there are usually irregularities, such impurities and defects, which can reduce the energybarrier of nematic–isotropic transition The phase transition takes place before the thermodynamicinstability limits (the supercooling or superheating temperature) Under an optical microscope, it isusually observed that with decreasing temperature nematic ‘islands’ are initiated by irregularities andgrow out of the isotropic ‘sea’, and with increasing temperature isotropic ‘lakes’ are produced byirregularities and grow on the nematic ‘land’ The irregularities are called nucleation seeds and thetransition is a nucleation process In summary, nematic–isotropic transition is a first-order transition andthe order parameter changes discontinuously, there is an energy barrier in the transition, and the

2.01.00.0–1.0–2.0

Figure 1.7 Free energy vs order parameter at various temperatures in Landau-de Gennes theory

transition is a nucleation process; superheating and supercooling occur In a second-order transition,there is no energy barrier and the transition occurs simultaneously everywhere at the transitiontemperature (the critical temperature).

There are a few points worth mentioning in Landau–de Gennes theory First, the theory works well

at temperatures near the transition temperature At temperatures far below the transition temperature,however, the order parameter increases without limit with decreasing temperature, and the theorydoes not work well because the maximum order parameter should be 1 In Figure 1.6, the parametersare chosen in such a way that the fitting is good for a relatively wide temperatures region,

TNI T¼ 2b2=9ac¼ 6:3C, which is much larger than the value (18C) measured by light-scatteringexperiments in the isotropic phase [18] There are fluctuations in orientational order in the isotropicphase, which results in a variation of refractive index in space and causes light scattering The intensity

of the scattered light is proportional to 1=ðT  TÞ

1.3.3 Maier–Saupe theory

In the nematic phase, there are interactions, such as the van der Waals interaction, between the liquidcrystal molecules Because the molecular polarizability along the long molecular axis is larger thanalong the short transverse molecular axis, the interaction is anisotropic and results in the parallelalignment of the rod-like molecules In the spirit of the mean field approximation, Maier and Saupeintroduced an effective single molecule potential V to describe the intermolecular interaction [19, 20].The potential has the following properties (1) It must be a minimum when the molecule orients along theliquid crystal director (the average direction of the long molecular axis of the molecules) (2) Its strength

is proportional to the order parameter S¼ hP2ðcos yÞi because the potential well is deep when themolecules are highly orientationally ordered and vanishes when the molecules are disordered (3) Itassures that the probabilities for the molecules pointing up and down are the same The potential inMaier–Saupe theory is given by

VðyÞ ¼ vS 3

2cos

2y12

(1.47)

where v is the orientational interaction constant of the order of 0:1 eV and y is the angle between thelong molecular axis and the liquid crystal director as shown in Figure 1.4 The probability f of themolecule orienting along the direction with polar angle y is governed by the Boltzmann distribution:

From the orientational distribution function we can calculate the order parameter:

S¼1

Z

Zp 0

P2ð cos yÞeVðyÞ=kB Tsin ydy¼1

Z

Zp 0

P2ð cos yÞevSP2 ðyÞ=k B Tsin ydy (1.50)

We introduce a normalized temperature t¼ kBT =v For a given value of t, the order parameter S can

be found by numerically solving Equation (1.50) An iteration method can be used for the numericalcalculation of the order parameter: (1) choose an initial value for the order parameter, (2) substitute it

into the right hand side of Equation (1.50), and (3) calculate the order parameter Use the newly obtainedorder parameter to repeat the above process until a stable value is obtained As shown in Figure 1.8(a),there are three solutions: S1, S2, and S3 In order to determine which is the actual solution, we have toexamine the corresponding free energies The free energy F has two parts, F¼ U  TEn, where U is theintermolecular interaction energy and Enis the entropy The single molecular potential describes theinteraction energy between one liquid crystal molecule and the remaining molecules of the system Theinteraction energy of the system with N molecules is given by

U¼1

2NhVi ¼ N

2Z

Zp 0

where the factor1avoids counting the intermolecular interaction twice The entropy is calculated byusing Equation (1.32):

En¼ NkBhln f i ¼ NkB

Z

Zp 0ln½ f ðyÞeVðyÞ=kB Tsin ydy (1.52)

From Equation (1.48) we have ln½ f ðyÞ ¼ V ðyÞ=kBT lnZ; therefore En ¼ ðN =T ÞhV i þ NkBln Zand the free energy is

Although the second term in this equation looks abnormal, this equation is correct and can be checked

by calculating the derivative of F with respect to S:

P2ð cos yÞevSP 2 ðyÞ=k B Tsin ydy

which is consistent with Equation (1.50) The free energies corresponding to the solutions are shown inFigure 1.8(b) The nematic–isotropic phase transition temperature is tNI ¼ 0:22019 For temperatureshigher than tNI, the isotropic phase with order parameter S¼ S1¼ 0 has a lower free energy and thus isstable For temperatures lower than tNI, the nematic phase with order parameter S¼ S2has a lower freeenergy and thus is stable The order parameter jumps from 0 to Sc¼ 0:4289 at the transition

In the Maier–Saupe theory there are no fitting parameters The predicted order parameter as a function

of temperature is universal, and agrees qualitatively, but not quantitatively, with experimental data Thisindicates that higher order terms are needed in the single molecule potential, i.e.,

VðyÞ ¼X

i

½vihPið cos yÞiPið cos yÞ (1.55)

where Piðcos yÞ ði ¼ 2; 4; 6; Þ are the ith-order Legendre polynomials The fitting parameters are vi.With higher order terms, better agreement with experimental results can be achieved

Maier–Saupe theory is very useful in considering liquid crystal systems consisting of more than onetype of molecule, such as mixtures of nematic liquid crystals and dichroic dyes The interactions betweendifferent molecules are different and the constituent molecules have different order parameters.All the theories discussed above do not predict well the orientational order parameter for temperaturesfar below TNI The order parameter as a function of temperature is better described by the empiricalformula [21]

where V and VNI are the molar volumes at T and TNI, respectively

1.4 Elastic Properties of Liquid Crystals

In the nematic phase, the liquid crystal director ~n is uniform in space in the ground state In reality, theliquid crystal director ~n may vary spatially because of confinement or external fields This spatialvariation of the director, called the deformation of the director, costs energy When the variation occursover a distance much larger than the molecular size, the orientational order parameter does not changeand the deformation can be described by a continuum theory analogous to the classic elastic theory of asolid The elastic energy is proportional to the square of the spatial variation rate

1.4.1 Elastic properties of nematic liquid crystals

There are three possible deformation modes of the liquid crystal director as shown in Figure 1.9 Wechoose the cylindrical coordinate such that the z axis is parallel to the director at the origin of thecoordinate: ~nð0Þ ¼ ˆz Consider the variation of the director at an infinitely small distance from the origin.When moving in the radial direction, there are two possible modes of variation: (1) the director tiltstoward the radial direction ˆr as shown in Figure 1.9(a), and (2) the director tilts toward the azimuthal

direction ˆf as shown in Figure 1.9(b) The first mode is called splay, where the director atðdr; f; z ¼ 0Þis

where K11 is the splay elastic constant

The second mode is called twist, where the director atðdr; f; z ¼ 0Þ is

~nðdr; f; z ¼ 0Þ ¼ dnfðdrÞ ˆfþ ½1 þ dnzðdrÞˆz (1.59)

where dnf 1 and dnz¼ ðdnfÞ2=2, a higher order term which can be neglected The spatial variationrate is @nf=@r and the corresponding elastic energy is

where K22 is the twist elastic constant

When moving in the z direction, there is only one possible mode of variation, as shown inFigure 1.9(c), which is called bend The director atðr ¼ 0; f; dzÞ is

(c)(b)

d r

Figure 1.9 The three possible deformations of the liquid crystal director: (a) splay; (b) twist; and(c) bend

Note that when r¼ 0, the azimuthal angle is not well defined and we can choose the coordinate suchthat the director tilts toward the radial direction The corresponding elastic energy is

where K33 is the bend elastic constant Because dnz is a higher order term, @nz=@z 0 and

@nz=@r 0 Recall that r  ~njr¼0; z¼0¼ ð1=rÞ@ðrnrÞ=@r þ ð1=rÞ@nf=@fþ @nz=@z¼ @nr=@rþ dnr.Because @nr=@r is finite and dnr 1, r  ~njr¼0; z¼0¼ @nr=@r The splay elastic energy can beexpressed as fsplay¼ ð1=2ÞK11ðr  ~nÞ2 Because ~n¼ ˆz, at the origin ~n njr¼0; z¼0 nÞz¼

@nf=@r The twist elastic energy can be expressed as ftwist¼ ð1=2ÞK22ð~n nÞ2 Because ~n

njr¼0; z¼0 nÞr nÞf¼ @nr=@z, the bend elastic energy can be expressed as

fbend ¼ ð1=2ÞK33ð~n nÞ2 Putting all the three terms together, we obtain the elastic energydensity:

fela¼1

2K11ðr  ~nÞ2þ1

2K22ð~n nÞ2þ1

This elastic energy is often referred to as the Oseen–Frank energy and K11, K22, and K33are referred

to as the Frank elastic constants because of his pioneering work on the elastic continuum theory of liquidcrystals [22] The value of the elastic constants can be estimated in the following way When a significantvariation of the director occurs in a length L, the angle between the average directions of the longmolecular axes of two neighboring molecules isða/LÞ, where a is the molecular size When the averagedirections of the long molecular axes of two neighboring molecules are parallel, the intermolecularinteraction energy between them is a minimum When the average direction of their long molecularaxes makes an angle ofða/LÞ, the intermolecular interaction energy increases to ða /LÞ2u, where u is theintermolecular interaction energy associated with orientation and is about 0:1 eV The increase of theinteraction energy is the elastic energy, i.e.,

aL

19 J

1 nm¼ 1011N

Experiments show that usually the bend elastic constant K33is the largest and the twist elastic constant

K22 is the smallest As an example, at room temperature the liquid crystal 5CB has these elastic

f24¼ K24r  ð~nr  ~nþ ~n nÞ, respectively [23] The volume integral of these two termscan be changed to a surface integral because of the Gauss theorem

1.4.2 Elastic properties of cholesteric liquid crystals

So far we have considered liquid crystals consisting of molecules with reflectional symmetry Themolecules are the same as their mirror images, and are called achiral molecules The liquid crystal 5CBshown in Figure 1.1(a) is an example of an achiral molecule Now we consider liquid crystals consisting

of molecules without reflectional symmetry The molecules are different from their mirror images andare called chiral molecules Such an example is CB15 shown in Figure 1.10(a) It can be regarded as ascrew, instead of a rod, in considering its physical properties After considering the symmetry where ~nand~n are equivalent, the generalized elastic energy density is

eta; then qo¼ 0 When the liquid crystal is in the ground state with minimum free energy,

fela¼ 0, which requires that r  ~n¼ 0, ~n n¼ 0, and ~n n¼ 0 This means that in theground state, the liquid crystal director ~n is uniformly aligned along one direction

If the liquid crystal molecule is chiral and thus has no reflectional symmetry, the system changes underreflectional symmetry operation The elastic energy may change It is no longer required that fela¼ feta0 ,and thus qomay not be zero When the liquid crystal is in the ground state with minimum free energy,

fela¼ 0, which requires that r  ~n¼ 0, ~n n¼ qo, and ~n n¼ 0 A director configurationwhich satisfies the above conditions is

nx¼ cosðqozÞ; ny¼ sinðqozÞ; nz¼ 0 (1.66)

and is schematically shown in Figure 1.11 The liquid crystal director twists in space This type of liquidcrystal is called a cholesteric liquid crystal The axis around which the director twists is called the helicalaxis and is chosen to be parallel to z here The distance Poover which the director twists by 360is calledthe pitch and is related to the chirality by

Figure 1.10 (a) Chemical structure of a typical chiral liquid crystal molecule; (b) physical model of achiral liquid crystal molecule

Depending on the chemical structure, the pitch of a cholesteric liquid crystal could take any value in theregion from a few tenths of a micron to infinitely long The periodicity of a cholesteric liquid crystal withpitch Po is Po=2 because ~n and ~n are equivalent Cholesteric liquid crystals are also called chiralnematic liquid crystals and denoted as N Nematic liquid crystals can be considered as a special case ofcholesteric liquid crystals with an infinitely long pitch.

In practice, a cholesteric liquid crystal is usually obtained by mixing a nematic host with a chiraldopant The pitch of the mixture is given by

where x is the concentration of the chiral dopant and (HTP is the helical twisting power of the chiraldopant, which is mainly determined by the chemical structure of the chiral dopant and depends onlyslightly on the nematic host

1.4.3 Elastic properties of smectic liquid crystals

Smectic liquid crystals possess partial positional orders besides the orientational order exhibited innematic and cholesteric liquid crystals Here we only consider the simplest case: smectic-A The elasticenergy of the deformation of the liquid crystal director in smectic-A is the same as in the nematic liquidcrystal In addition, the dilatation (compression) of the smectic layer also costs energy which is given by[24]

thickness of the smectic layer, respectively Typical values of B are about 106107J=m3, which are

103to 104smaller than those in a solid In a slightly deformed smectic-A liquid crystal, we consider aclosed loop as shown in Figure 1.12 The total number of layers traversed by the loop is zero, whichcan be mathematically expressed asH

~n dl ¼ 0 Using the Stokes theorem, we haveR

n d~s¼H

~n dl ¼ 0 Therefore in smectic-A we have

which assures that ~n n¼ 0 and ~n n¼ 0 The consequence is that twist and benddeformations of the director are not allowed (because they change the layer thickness and cost toomuch energy) The elastic energy in a smectic-A liquid crystal is

to thermal fluctuations This effect causes the pitch of the cholesteric liquid crystal to increase withdecreasing temperature and diverge at the transition temperature as shown in Figure 1.13 As will bediscussed later, a cholesteric liquid crystal with pitch P exhibits Bragg reflection at the wavelength

l¼ nP, where n is the average refractive index of the material If l ¼ nP is in the visible light region,

n

Figure 1.12 Schematic diagram showing the deformation of the liquid crystal director and thesmectic layer in the smectic-A liquid crystal

T P

IsotropicCholesteric

Smectic-A

Visible region

2

T T1

Figure 1.13 Schematic diagram showing how the pitch of a thermochromic cholesteric liquid crystalchanges

the liquid crystal reflects colored light When the temperature is varied, the color of the liquid crystalchanges These types of cholesteric liquid crystals are known as thermochromic cholesteric liquidcrystals [24] As shown in Figure 1.13, the reflected light is in the visible region for temperatures inthe region from T1 to T2 There are liquid crystals with DT¼ T1 T2¼ 1 If there are twothermochromic cholesteric liquid crystals with different cholesteric–smectic-A transition temperatures,mixtures with different concentrations of the two components will exhibit color reflections at differenttemperatures This is how thermochromic cholesteric liquid crystals are used to make thermometers.

1.5 Response of Liquid Crystals to Electromagnetic Fields

Liquid crystals are anisotropic dielectric and diamagnetic media [1,25] Their resistivities are very high

ð  1010O cmÞ Dipole moments are induced in them by external fields They have different dielectricpermittivities and magnetic susceptibilities along the directions parallel and perpendicular to the liquidcrystal director

1.5.1 Magnetic susceptibility

We first consider magnetic susceptibility Because the magnetic interaction between the molecules isweak, the local magnetic field of the molecules is approximately the same as the externally appliedmagnetic field For a uniaxial liquid crystal, a molecule can be regarded as a cylinder When a magneticfield ~H is applied to the liquid crystal, it has different responses to the applied field, depending on theangle between the long molecular axis ~a and the field ~H The magnetic field can be decomposed into aparallel component and a perpendicular component as shown in Figure 1.14 The magnetization ~M isgiven by

~

M¼ N

k?þ Dkaxax Dkaxay DkaxazDkayax k?þ Dkayay DkayazDkazax Dkazay k? þ Dkazaz

0B

1C

A  ~H¼ N k$ ~H (1.73)

where aiði ¼ x; y; zÞ are the projections of ~a in the x, y, and z directions in the lab frame whose z axis isparallel to the liquid crystal director: az¼ cosy, ax¼ sin y cos f, and ay¼ sin y sin f The moleculeswivels because of thermal motion The averaged magnetization is ~M¼ Nh k$i  ~H For a uniaxialliquid crystal, recall that hcos2yi ¼ ð2S þ 1Þ=3, hsin2yi ¼ ð2  2SÞ=3, hsin2fi ¼ hcos2fi ¼ 1=2,andhsin f cos fi ¼ 0 Therefore

1